Theorem dfdm3 4798

Description: Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)

Assertion

Ref	Expression
dfdm3	|- dom A = { x | E. y <. x , y >. e. A }

Distinct variable group:   x, y, A

Proof of Theorem dfdm3

Step	Hyp	Ref	Expression
1		df-dm 4621	. 2 |- dom A = { x | E. y x A y }
2		df-br 3990	. . . 4 |- ( x A y <-> <. x , y >. e. A )
3	2	exbii 1598	. . 3 |- ( E. y x A y <-> E. y <. x , y >. e. A )
4	3	abbii 2286	. 2 |- { x | E. y x A y } = { x | E. y <. x , y >. e. A }
5	1, 4	eqtri 2191	1 |- dom A = { x | E. y <. x , y >. e. A }

Syntax hints:   = wceq 1348   E.wex 1485   e. wcel 2141   {cab 2156   <.cop 3586   class class class wbr 3989   dom cdm 4611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-br 3990  df-dm 4621

This theorem is referenced by:  csbdmg  4805